1051
ISSN 1472-2739 (on-line) 1472-2747 (printed)
Algebraic & Geometric Topology
ATG
Volume 3 (2003) 1051–1078
Published: 16 October 2003
Resolutions of p-stratifolds with
isolated singularities
Anna Grinberg
Abstract Recently M. Kreck introduced a class of stratified spaces called
p-stratifolds [Kr3]. He defined and investigated resolutions of p-stratifolds
analogously to resolutions of algebraic varieties. In this note we study a very
special case of resolutions, so called optimal resolutions, for p-stratifolds
with isolated singularities. We give necessary and sufficient conditions for
existence and analyze their classification.
AMS Classification 58A32; 58K60
Keywords Stratifold, stratified space, resolution, isolated singularity
1
Introduction
Roughly speaking, p-stratifolds are topological spaces which are constructed
by attaching manifolds with boundary by a map to the already inductively
constructed space. The attaching map has to fulfill some subtle properties.
There is a more general notion of stratifolds introduced by M. Kreck [Kr3].
However, the only results concerning the resolution of stratifolds exist after
going over to the subclass of p-stratifolds.
The situation simplifies very much, if we consider only p-stratifolds with isolated
singularities, where the construction is done in two steps only. The first step
is the choice of a countable number of points {xi }i∈I⊆N which will become
the isolated singularities. The second step is the choice of a smooth manifold
N of dimension m, together with a proper map g : ∂N −→ {xi }i∈I , where
{xi }i∈I is considered as 0-dimensional manifold and the collection of boundary
components f −1 (xi ) is equipped with a germ of collars. The p-stratifold is
obtained by forming
= N ∪g {xi }i∈I .
S
We reformulate this in a slightly different way.
Definition An m-dimensional p-stratifold with isolated singularities is a topological space
together with a proper map f : N −→ , where
S
c Geometry & Topology Publications
S
1052
Anna Grinberg
• N is an m-dimensional manifold with boundary,
• f |N̊ is a homeomorphism onto its image,
•
S − f (N̊) is a discrete countable set, denoted by Σ, the singular set,
• f −1 (x) is equipped with a germ of collars for all x ∈ Σ,
• U ⊂
S
is open if and only if U ∩ Σ is open and f −1 (U ) is open in N .
S
The manifold f (N̊) is called the top stratum, Σ is called the 0-stratum of
.
Choose an identification Σ = {xi }i∈I⊆N and denote the collection of boundary
components mapped to a singular point Li := f −1 (xi ) the link of
at xi ∈ Σ.
S
A collar around Li is a diffeomorphism ci : Li × [0, i ) −→ Ui , where Ui is an
open neighbourhood of Li in N and i > 0, such that ci |Li ×{0} is the identity
map on Li . The germ is an equivalence class of collars, where two collars
ci : Li × [0, i ) −→ Ui and c̃i : Li × [0, ˜i ) −→ Ũi are called equivalent if there
is a positive δ ≤ min{i , ˜i }, such that ci |Li ×[0,δ) ≡ c̃i |Li ×[0,δ) . The role of the
collars becomes clear if we define smooth maps from a smooth manifold to a
p-stratifold.
S
S
Definition Let
be a p-stratifold with isolated singularities and ρ :
−→ R
a continuous map. The map ρ is called smooth if ρf |N̊ : N̊ −→ R is smooth and
there are representatives of the germ of collars ci : Li × [0, εi ) −→ N satisfying
for all i:
ρf (ci (x, t)) = ρf (x) for all x ∈ Li .
S
Let M be a smooth manifold. A continuous map g : M −→
is called smooth,
if for all smooth maps ρ :
−→ R the composition ρg is again smooth.
S
It is not hard to verify that the map g is smooth if and only if the restriction
g|g−1 (S −Σ) is smooth.
The most important examples of p-stratifolds as defined above are algebraic
varieties with isolated singularities.
Example (Algebraic varieties with isolated singularities)
Consider an algebraic variety V ⊂ Rn with isolated singularities, i.e. the singular set Σ is zero-dimensional. Let si ∈ Σ be a singular point. There is nothing
to do if si is open in V . Otherwise consider the distance function ρi on Rn
given by ρi (x) := ||x − si ||2 . It is well known that there is an εi > 0 such
that on Vεi (si ) := V ∩ Dεi (si ) the restriction ρi |Vεi −{si } has no critical values.
Here Dεi (si ) denotes the closed ball in Rn of radius εi centered at si . Set
∂Vεi (si ) := Vεi (si ) ∩ ∂Dεi (si ).
Algebraic & Geometric Topology, Volume 3 (2003)
1053
Resolutions of p-stratifolds with isolated singularities
By following the integral curves of the gradient vector field of ρi |Vεi −{si } , we
obtain a diffeomorphism
h : ∂Vεi (si ) × [0, i )
/
PP
PP
PP
PP
pr2 PPPP
P
Vεi − {si }
ss
ss
ss
s
s i −ρi
ss
y
(
[0, i )
being the identity on ∂Vεi (si ) × {0}, see [H, §6.2]. We extend this map to a
continuous map
h̄ : ∂Vεi (si ) × [0, i ] −→ Vεi .
Finally, we define the manifold N (with obvious collar) by setting
N := V − (ti D̊εi (si )) ∪id ∂Vεi (si ) × [0, i ].
The map f = id ∪ h̄ : N −→ V gives V the structure of a p-stratifold with
isolated singularities.
Since every complex algebraic variety is in particular a real one, we obtain the
same result for a complex algebraic variety with isolated singularities.
From now on all p-stratifolds are p-stratifolds with isolated singularities. To
simplify the notation combine the representatives of the collars ci : Li × [0, εi )
to a single map c : ti Li × [0, εi ) −→ N . Using this map the singular set Σ is
equipped with the germ of neighbourhoods [U Σ] by taking U Σ := f (im c) t
(Σ − f (∂N )). The collars also give us a retraction r : U Σ −→ Σ.
We also introduce the germ of closed neighbourhoods [U Σ] by setting U Σ :=
f (c(ti Li × [0, εi /2])) t (Σ − f (∂N )). If we want to make the dependency on the
representative of the germ of collars clear, we sometimes write U Σc and U Σc .
S be an n-dimensional p-stratifold with top stratum f (N̊ ).
S is a proper map p : Sˆ −→ S such that
Definition Let
A resolution of
•
Sˆ is a smooth manifold;
• p is a proper smooth map;
• the restriction of p on p−1 (f (N̊ )) is a diffeomorphism on f (N̊);
• p−1 (f (N̊ )) is dense in
Sˆ;
• the inclusion Σ̂ := p−1 (Σ) ,→ U Σ̂ := p−1 (U Σ) is a homotopy equivalence
for a representative of the neighbourhood U Σ of Σ.
Algebraic & Geometric Topology, Volume 3 (2003)
1054
Anna Grinberg
S
S
A resolution p : ˆ −→
is called optimal, if p|Σ̂ : Σ̂ −→ Σ is an [n/2]equivalence. In particular, it follows that p : ˆ −→
is an [n/2]-equivalence
as well.
S
S
If the manifold N is equipped with more structure, e.g. orientation or spinstructure, we introduce corresponding resolutions, which have more structure.
S
Definition Let
= f (N ) ∪ {xi }i be a p-stratifold with oriented N . A
resolution p : ˆ −→
is called an oriented resolution, if ˆ is oriented and
p|p−1 (f (N̊ )) is orientation preserving. Analogously, if N is spin, then p : ˆ −→
is called a spin resolution if ˆ is spin and p| −1
preserves the spin
S
S
S
S
S
structure.
p
S
(f (N̊ ))
If V is an algebraic variety, Hironaka has shown [Hi] that there is a resolution of
singularities in the sense of algebraic geometry. The above topological definition
is modelled on the one from algebraic geometry. All conditions are analogous
except the last one, which is always fulfilled in the context of algebraic geometry.
As explained in [BR], a neighbourhood U of the singular set Σ of an algebraic
variety V such that the inclusion Σ ,→ U is a homotopy equivalence can be
obtained from a proper algebraic map ρ : V −→ R with Σ = ρ−1 (0) by
taking U = ρ−1 [0, r), provided r > 0 is small enough. Thus for a resolution
p : V̂ −→ V the preimage Û := p−1 (U ) is a neighbourhood of Σ̂ := p−1 (Σ) in V̂
obtained from ρ̂ := ρp, hence the inclusion Σ̂ ,→ Û is a homotopy equivalence.
Note that U Σ̂ is a smooth manifold with boundary diffeomorphic to ∂N . Consider the preimage of the neighbourhood of each singularity and set U Σ̂i :=
p−1 (f ci (Li × [0, εi /2])), where ci is the restriction of the collar to Li × [0, εi /2].
S
It is not hard to verify that a resolution p : ˆ −→
the manifolds U Σ̂i are ([n/2] − 1)-connected.
S
is optimal if and only if
In contrast to algebraic varieties, resolutions of stratifolds in general do not
exist, not even for isolated singularities. But in this case there is a simple
necessary and sufficient condition, see [Kr3] and §6 for a proof.
Theorem 1 An n-dimensional p-stratifold with isolated singularities admits a
resolution if and only if each link of the singularity Li , vanishes in the bordism
group Ωn−1 .
S
Example The p-stratifold
= CP 2 × I ∪f {x0 , x1 } with the obvious strati2
fication, such that f (CP × {0}) = x0 and f (CP 2 × {1}) = x1 , does not admit
a resolution.
Algebraic & Geometric Topology, Volume 3 (2003)
1055
Resolutions of p-stratifolds with isolated singularities
To give a feeling of the result concerning optimal resolution, we formulate the
following special case which will be derived as Corollary 7 of Theorem 5 (cf.
§2).
S
Corollary Let
be a p-stratifold with parallelizable links of singularities Li .
Assume Li is bounded by a parallelizable manifold, then
admits an optimal
resolution.
S
We have shown above that every algebraic variety with isolated singularities
admits a structure of a p-stratifold. One may ask the converse question. When
does a p-stratifold with isolated singularities admit an algebraic structure? The
following Theorem of Akbulut and King [AK, Thm. 4.1] clarifies the situation
in the case of a real algebraic structure.
Theorem 2 A topological space X is homeomorphic to a real algebraic set
with isolated singularities if and only if X is obtained by taking a smooth
compact manifold M with boundary ∂M = ∪ri=1 Li , where each Li bounds,
then crushing some Li ’s to points and deleting the remaining Li ’s.
Combining this result with Theorem 1 we immediately obtain:
S
Corollary 3 A compact p-stratifold
with isolated singularities is homeomorphic to a real algebraic set with isolated singularities if and only if
admits a resolution.
S
Example (Resolutions of hypersurfaces with isolated singularities)
Let p : Rn+1 −→ R be a polynomial with isolated singularities {si }i , i.e.
si ∈ V := p−1 (0) and si is an isolated critical point of p. Assume further that
the points si are not open. According to a previous example, the hypersurface
V admits a canonical structure of a p-stratifold. We have to investigate the
link of the singularity, which is given by ∂Vεi (si ).
Choose a δ > 0 such that all c with |c| ≤ δ are regular values of p and take c
such that p−1 (c) 6= ∅. Then p−1 (c) is a smooth manifold with trivial normal
bundle. With the help of the gradient vector field we see that p−1 (c) ∩ Sεni (si ) is
diffeomorphic to p−1 (0)∩Sεni (si ) = ∂Vεi (si ). Thus, ∂Vεi (si ) ∼
= p−1 (c)∩Sεni (si ) ∼
=
−1
n+1
∂(p (c) ∩ Dεi (si )). We see that a resolution always exists, and since the
bounding manifolds are automatically parallelizable, we even obtain an optimal
resolution after choosing an appropriate bordism (compare with Figure 1).
Algebraic & Geometric Topology, Volume 3 (2003)
1056
Anna Grinberg
V̂1 = p−1 (−ε)
(not optimal)
V = p−1 (0)
V̂2 = p−1 (ε)
(optimal)
Figure 1: p(x, y, z) = x2 + y 2 − z 2
In the case of a complex polynomial p : Cn+1 −→ C (n > 0), every deformation
p−1 (c) gives us an optimal resolution, provided ||c|| is small enough. This
follows from a result of Milnor [Mi4, Thm. 6.5] which states that M := p−1 (c)∩
Dε2n+2 (si ) is homotopy equivalent to a wedge of µ ≥ 1 copies of S n and thus
(n − 1)-connected.
Consider another interesting class of p-stratifolds with isolated singularities,
namely those arising from a smooth group action.
Definition A smooth S 1 -action on a smooth manifold M is called semi-free if
the action is free outside of the fixed point set, i.e. if gx = x for a g ∈ S 1 , g 6= 1
and x ∈ M , then hx = x for all h ∈ S 1 .
Lemma 4 Let M be a closed oriented manifold with semi-free S 1 -action with
only isolated fixed points. Then M/S 1 admits an oriented resolution if and
only if dim M ≡ 0(mod 4).
Proof Let dim M = n. There is nothing to show if the action is free. Thus let
x ∈ M be a fixed point. The differential of the action gives a representation of
S 1 on Tx M and there is an equivariant local diffeomorphism from Tx M onto a
neighbourhood of x in M . According to [BT, Prop. (II.8.1)], every irreducible
Algebraic & Geometric Topology, Volume 3 (2003)
1057
Resolutions of p-stratifolds with isolated singularities
S 1 on Rn is equivalent to:

 n
...
...
0
z 1

.. 
..
 0
.
.


 ..
.. 
..

 .
.
·
.


 ..
..
n
[n/2]

 .
. z
0
0
... ...
0
1
representation of
 n
z 1 0

 0
·

 ..
.
..
 .

 ..
 .
0
considered as a representation on
C × · · · × C × R, if n is odd
0
·
..
.
...
...
..
.
...
·
..
.
..
...
.
·
0
0
..
.
..
.








0 
z nn/2
considered as a representation on
C × · · · × C, if n is even
Since the action is semi-free and x an isolated fixed point we conclude that
dim M is even. We can further assume ni = 1 for all i ∈ {1, . . . , n/2}. Let
dim M = 2m and let {x1 , . . . , xk } be the set of fixed points. Choose equivariant
disks Dxi around xi . In this situation we have
M/S 1 = (M − ti D̊xi )/S 1 ∪ {x1 , . . . , xk }.
The domain of the top stratum is then given by N := (M − ti D̊xi )/S 1 and
the singular set is Σ := {x1 , . . . , xk }. The links of singularities are given by
Li ∼
= S 2m−1 /S 1 = CP m−1 . Using Theorem 1 we conclude that the resolution
exists if and only if [CP m−1 ] vanishes in ΩSO
2m−2 . For m = 2l + 1 the signature
m−1
m−1
of CP
is equal to 1, hence CP
does not bound. In the case of an even
m = 2l + 2 we have CP 2l+1 = S 4l+3 /S 1 = S(Hl+1 )/S 1 , where S(Hl+1 )/S 1 is
the sphere bundle
S 2 = S 3 /S 1

/
S(Hl+1 )/S 1
S(Hl+1 )/S 3
= HP l
and the associated disk bundle bounds.
As mentioned before, we are particularly interested in the classification of resolutions. Thus we have to decide when we are going to consider two resolutions
as equivalent. We can restrict our attention to the resolving manifolds and
introduce a relation on them, e.g. diffeomorphism, but in this case we completely ignore an important part of the resolution data, namely the resolving
map. Hence, one can ask for diffeomorphisms between the resolving manifolds
commuting with the resolving maps. This relation is very strong and, therefore,
very hard to control. In the following definition, we combine these two ideas.
Algebraic & Geometric Topology, Volume 3 (2003)
1058
Anna Grinberg
S
S
S
S
S
Definition Let
be a p-stratifold and p : ˆ −→
and p0 : ˆ0 −→
two
resolutions of . We call the resolutions equivalent, if, for every representative
of the neighbourhood germ U Σc , there is a diffeomorphism ϕc : ˆ −→ ˆ0
such that the following holds:
S
S
• p0 ϕc = p on
S
Sˆ − U Σ̂c and
• rp0 ϕc = rp on U Σ̂c , where r : U Σc −→ Σ is the neighbourhood’s retrac-
tion.
This means outside of an arbitrary small neighbourhood of the singularity, the
diffeomorphism commutes with the resolving maps and near Σ it only commutes
after the composition with the retraction.
Observe that ϕc gives a diffeomorphism ∂U Σ̂c −→ ∂U Σ̂0c .
The classification of optimal resolutions is quite a difficult problem. For if
Sˆ is an optimal resolution of S , then Sˆ]S is again optimal for an arbitrary
homotopy sphere S . In particular, consider the sphere S n stratified as D n ∪ pt,
then every homotopy sphere S n gives us a resolution of S n . Thus, we weaken
the problem and ask for the equivalence up to a homotopy sphere.
S
S
S
S
Definition Two resolutions ˆ −→
and ˆ0 −→
are called almost
0
ˆ
ˆ
equivalent if ]S is equivalent to
for a homotopy sphere S .
S
S
A special case of the classification result is the following.
S
S
S
Ŝ
S
Corollary Let ˆ −→
and 0 −→
be two resolutions of a 2n-dimensional p-stratifold
having (n − 2)-connected links of isolated singularities.
Assume that n ≡ 6 (mod 8)and that U Σ̂i and U Σ̂0i are parallelizable with
compatible parallelizations on the boundary. Let further e(U Σ̂i ) = e(U Σ̂0i ) and
sign(U Σ̂i ∪∂ U Σ̂0i ) = 0. Then there is a k ∈ {0, 1} such that ˆ]k(S n × S n ) is
almost equivalent to ˆ0 ]k(S n × S n ).
S
2
S
Existence of optimal resolutions
Before proceeding with the existence of an optimal resolution we need to introduce some notation. For a topological space X let Xhki be the k -connected
cover of X , which always comes with a fibration p : Xhki −→ X . For further
information see for example [Ba]. We take X to be the classifying space BO
BOhki
and denote by Ωn
the bordism group of closed n-dimensional manifolds
together with a lift of the normal Gauss map, compare [St, Chap. I].
Algebraic & Geometric Topology, Volume 3 (2003)
1059
Resolutions of p-stratifolds with isolated singularities
Theorem 5 An n-dimensional p-stratifold with isolated singularities admits
an optimal resolution if and only if the normal Gauss map νj : Lj −→ BO
BOh[n/2]−1i
admits a lift over BOh[n/2] − 1i, such that [Lj , ν̄j ] = 0 in Ωn−1
.
S
S
Proof Let
be a p-stratifold with isolated singularities {xi }i∈I and ˆ an
optimal resolution of
. Set Wi := U Σ̂i , then Wi is a smooth manifold with
boundary Li for i ∈ I ⊆ N. Consider the normal Gauss map, together with
the ([n/2] − 1)-connected cover over BO.
S
BOh[n/2] − 1i
ν̄j
o
o
o
o
o
o
νj : W j
_
o
?
F
7
/
BO
The obstructions for the existence of a lift lie in H̃ r (Wj , πr−1 (F )). Note that we
can use global coefficients since the fibration BOh[n/2] − 1i −→ BO is simple.
Since the resolution is optimal the manifold Wi is ([n/2] − 1)-connected, hence
H̃ r (Wi ) = 0 for r < [n/2].
Using the properties of the connected cover it follows from the long exact homotopy sequence that πr (F ) = 0 for r ≥ [n/2] − 1.
Hence, there are no obstructions for the lifting of the normal Gauss map, thus
BOh[n/2]−1i
[Lj , νj ] vanishes in Ωn−1
.
The fact that the condition is also sufficient is an immediate consequence of the
following result from [Kr1, Prop. 4].
Theorem 6 Let ξ : B −→ BO be a fibration and assume that B is connected
and has a finite [m/2]-skeleton. Let ν̄ : M −→ B be a lift of the normal Gauss
map of an m-dimensional compact manifold M . Then if m ≥ 4, by a finite
sequence of surgeries (M, ν̄) can be replaced by (M 0 , ν̄ 0 ) so that ν̄ 0 : M 0 −→ B
is an [m/2]-equivalence.
For example, we obtain the following:
S
Corollary 7 Let
be a p-stratifold with parallelizable links of singularities
Li . Assume Li is bounded by a parallelizable manifold, then
admits an
optimal resolution.
Algebraic & Geometric Topology, Volume 3 (2003)
S
1060
3
Anna Grinberg
Classification of almost equivalent resolutions
Now we turn to the main result of this note. In this section we consider pstratifolds with isolated singularities of dimension 2n > 4 having (n − 2)connected links of singularities. The classification is based on the following
result from [Kr1, Thm. 2].
Theorem 8 For n > 2 let W1 and W2 be two compact connected 2nmanifolds with normal (n − 1)-smoothings in a fibration B . Let g : ∂W1 −→
∂W2 be a diffeomorphism compatible with the normal (n − 1)-smoothings ν1
and ν2 . Let further
• e(W1 ) = e(W2 ),
• [W1 ∪g (−W2 ), ν̄1 ∪ ν̄2 ] = 0 ∈ Ω2n (B).
Then g can be extended to a diffeomorphism G : W1 ]k(S n ×S n ) −→ W2 ]k(S n ×
S n ) for k ∈ N. Moreover, if B is 1-connected then k ∈ {0, 1}.
If W1 is simply connected and n is odd, then k can be chosen as 0, i.e. we
obtain diffeomorphism instead of stable diffeomorphism.
We have to explain some terms appearing in the last theorem. Let B be a
fibration over BO, a normal B -structure on a manifold M is a lift ν̄ of the
normal Gauss map ν : M −→ BO to B .
Definition Let B be a fibration over BO.
(1) A normal B -structure ν̄ : M −→ B of a manifold M in B is a normal
k -smoothing, if it is a (k + 1)-equivalence.
(2) We say that B is k -universal if the fiber of the map B −→ BO is connected and its homotopy groups vanish in dimension ≥ k + 1.
Obstruction theory implies that if B and B 0 are both k -universal and admit a
normal k -smoothing of the same manifold M , then the two fibrations are fiber
homotopy equivalent. Furthermore, the theory of Moore-Postnikov decompositions implies that for each manifold M there is a k -universal fibration B k
over BO admitting a normal k -smoothing, compare [Ba, §5.2]. Thus, the fiber
homotopy type of the fibration B k over BO is an invariant of the manifold M
and we call it the normal k -type of M .
There is an obvious bordism relation on closed n-dimensional manifolds with
normal B -structures and the corresponding bordism group is denoted Ωn (B).
Algebraic & Geometric Topology, Volume 3 (2003)
1061
Resolutions of p-stratifolds with isolated singularities
Applying the theorem to our situation, we first have to determine the normal
(n − 1)-type of an (n − 1)-connected 2n manifold.
Consider a subgroup H of G := πn (BO). Since the last group is always cyclic,
the group H is determined by an integer k , such that H = hkxi where x is
the generator of πn (BO). We call this integer the index of H .
Every subgroup H of G = πn (BO) gives us a fibration
B
/
P (K(G/H, n))
p
BOhn − 1i
/
θ
K(G/H, n)
where the map θ corresponds to the canonical epimorphism G −→ G/H . We
denote the space B belonging to the index-k group Bk . The composition
p
pk : Bk −→ BOhn − 1i −→ BO gives us a fibration over BO with fiber Fk .
Definition A 2n-dimensional manifold M is said to have the index k , if
ν∗ (πn (M )) is a subgroup of index k in πn (BO).
Theorem 9 The fibration Bk −→ BO is the normal (n−1)-type of an (n−1)connected 2n-dimensional manifold M , if and only if M is of index k .
The proof can be found in §6, which also contains proofs of the following two
theorems.
Now we look for conditions implying an (n − 1)-connected 2n-manifold to
be bordant to a homotopy sphere. Note first that as an easy consequence
from the universal coefficient theorem the first non-trivial homology group is
free. The homological information of M is stored in the triple (Hn (M ), Λ, ν)
where Λ denotes the intersection product Λ : Hn (M ) −→ Z, we often simply
write x · y for Λ(x, y). The last data ν is the normal bundle information,
described in the following way. According to a theorem of Haefliger [Hae]
every element of Hn (M ) is represented by an embedding S n ,→ M , and two
embeddings corresponding to the same homotopy class are regular homotopic.
Thus, assigning to an embedded sphere its normal bundle gives us a well defined
map ν : Hn (M ) −→ πn−1 (SO(n)).
Definition An (n − 1)-connected 2n-dimensional manifold M is called elementary if Hn (M ) admits a Lagrangian L w.r.t. Λ, such that ν|L ≡ 0.
Algebraic & Geometric Topology, Volume 3 (2003)
1062
Anna Grinberg
Theorem 10 Let M be an (n−1)-connected manifold of dimension 2n. Then
M is bordant to a homotopy sphere if and only if M is elementary.
Our main result, based on the last two theorems is:
S
S
Ŝ
S
0 −→
Theorem 11 For n > 2 let ˆ −→
and
be two optimal resolutions of a 2n-dimensional p-stratifold
with isolated singularities {xi }i∈I ,
such that each link Li is (n − 2)-connected. Assume further that for a suitable
representative U Σ the following conditions hold for all i ∈ I :
S
• e(U Σ̂i ) = e(U Σ̂0i );
• U Σ̂i and U Σ̂0i have the same index ki ;
• there exits normal (n − 1)-smoothings ν̄i and ν̄i0 of U Σ̂i and U Σ̂0i in the
fibration Bki −→ BO, such that νi |∂U Σ̂i = νi0 |∂U Σ̂0 ;
i
•
U Σ̂i ∪∂ U Σ̂0i
is elementary.
Sˆ is almost equivalent to Ŝ 0 .
If n is even, then Sˆ]k(S n × S n ) is almost equivalent to Ŝ 0 ]k(S n × S n ) for a
If n is odd, then
k ∈ {0, 1}.
4
Algebraic invariants
In this section we will find algebraic invariants, which allow us to decide whether
an (n − 1)-connected closed 2n-dimensional manifold is elementary or not (n >
2). Some proofs can be found in §6.7.
Recall the algebraic data corresponding to such a manifold M . We have a
triple (H, Λ, ν∗ ), where H = Hn (M ) is a free Z-module, Λ : H × H −→ Z is
the intersection product and ν∗ : H −→ πn−1 (SOn ) is a normal bundle map,
described in the previous section. The map ν∗ is not a homomorphism, but
satisfies the following equation:
ν∗ (x + y) = ν∗ (x) + ν∗ (y) + ∂Λ(x, y),
(∗)
where ∂ : Z ∼
= πn (S n ) −→ πn−1 (SOn ) is the boundary map from the long exact
homotopy sequence of the fibration SO(n) ,→ SO(n + 1) −→ S n , see [W1].
Thus, we obtain an algebraic object, the set Tn of triples (H, Λ, ν∗ ), where H is
a free Z-module, Λ : H ×H −→ Z is an (−1)n -symmetric unimodular quadratic
form and ν∗ : H −→ πn−1 (SOn ) is a map satisfying (∗). We want to investigate
Algebraic & Geometric Topology, Volume 3 (2003)
Resolutions of p-stratifolds with isolated singularities
1063
the assumptions under which an element (H, Λ, ν∗ ) ∈ Tn is elementary, i. e.
when H possesses a Lagrangian L with respect to Λ such that ν∗ |L ≡ 0.
We begin with an observation that for a 4k -dimensional manifold, the normal
bundle information can be replaced by the stable normal bundle map.
Lemma 12 Let n be even and let S n ,→ M 2n be an embedding. The normal
bundle ν(S n ) of S n in M is trivial if and only if ν ⊕ R is trivial and the Euler
class of ν(S n ) vanishes.
Thus, instead of considering (H, Λ, ν∗ ) ∈ Tn we can go over to (H, Λ, sν∗ ), where
sν∗ : H −→ πn−1 (SO) corresponds to the stable normal bundle map. Since the
Euler class of an embedded sphere representing x ∈ H can be identified with
the self intersection class we conclude:
Lemma 13 Let n be even. Then (H, Λ, ν∗ ) ∈ Tn is elementary if and only if
(H, Λ, sν∗ ) is elementary.
Let Tns denote the set of triples (H, Λ, sν∗ ), with H and Λ as above and
sν∗ : H −→ πn−1 (SO) a homomorphism. According to the different possibilities
for πn−1 (SO) we distinguish 3 cases.
(1) πn−1 (SO) = 0.
Claim (H, Λ, sν∗ ) ∈ Tns is elementary if and only if sign(Λ) = 0, where sign
denotes the signature of a quadratic form.
∼
=
(2) πn−1 (SO) = Z. Since Λ is unimodular it induces an isomorphism H −→
H ∗ , which we also denote by Λ. The map sν∗ gives an element of H ∗ and we
consider κsν∗ := Λ−1 (sν∗ ) ∈ H .
Claim (H, Λ, sν∗ ) ∈ Tns is elementary if and only if sign(Λ) = 0 and κ2sν∗ = 0.
(3) πn−1 (SO) = Z2 . Let (H, Λ, sν∗ ) be an element of Tns with vanishing
signature and suppose Λ is of type II , i.e. Λ(x, x) = 0 (mod 2) for all x ∈ H .
Note that since n 6= 8 in this case, an elementary element corresponding to a
manifold always has a type II quadratic form. Thus, the dimension of H is even
and according to [Mi1, Lem. 9] we can choose a basis {λ1 , . . . , λk , µ1 , . . . , µk }
satisfying
Λ(λi , λj ) = 0, Λ(µi , µj ) = 0 and Λ(λi , µj ) = δij .
Consider the set of all elements x ∈ H with Λ(x, x) = 0 and denote its image under canonical projection on H ⊗ Z2 by H 0 . The class Φ(H, Λ, sν∗ ) :=
Algebraic & Geometric Topology, Volume 3 (2003)
1064
Anna Grinberg
Pk
i=1 sν∗ (λi )sν∗ (µi ) ∈ Z2 is well-defined and is equal to the value sν∗ takes
most frequently on the finite set H 0 , the class is called Arf invariant.
Claim An element (H, Λ, sν∗ ) ∈ Tns with type II form Λ is elementary if and
only if sign(Λ) = 0 and Φ(H, Λ, sν∗ ) = 0.
Consider now the case of an odd n. The quadratic form is now skew symmetric.
Depending on the values of ν∗ there are again three different cases (compare
[Ke]), which were completely investigated in [W1].
(4) πn−1 (SOn ) = 0. In this case every element of Tn is elementary.
(5)
Pk πn−1 (SOn ) = Z2 . As in (3), we can define the Arf invariant Φ(H, Λ, ν∗ ) =
i=1 ν∗ (λi )ν∗ (µi ) ∈ Z2 , using a symplectic basis {λ1 , . . . , λk , µ1 , . . . , µk } of H .
Claim An element (H, Λ, ν∗ ) ∈ Tn is elementary if and only if Φ(H, Λ, ν∗ ) = 0.
(6) πn−1 (SOn ) = Z2 ⊕ Z2 . We consider again the stable normal bundle map
sν∗ : H −→ Z2 , the projection on the first component. As in (2) using Λ, we
obtain an element κ (determined mod 2H ) with sν∗ (x) = Λ(κ, x) (mod 2) for
all x ∈ H .
Claim An element (H, Λ, ν∗ ) ∈ Tn is elementary if and only if Φ(H, Λ, ν∗ ) = 0
and pr2 ν∗ (κ) = 0, where pr2 denotes the projection on the second component.
Knowing the algebraic description of elementary manifolds, we formulate a
special case of Theorem 11.
S
S
Ŝ
S
0 −→
Corollary 14 Let ˆ −→
and
be two resolutions of a 2ndimensional p-stratifold
having (n − 2)-connected links of isolated singularities. Assume that n ≡ 6 (mod 8)and that U Σ̂i and U Σ̂0i are parallelizable with
compatible parallelizations on the boundary. Let further e(U Σ̂i ) = e(U Σ̂0i ) and
sign(U Σ̂i ∪∂ U Σ̂0i ) = 0. Then there is a k ∈ {0, 1} such that ˆ]k(S n × S n ) is
almost equivalent to ˆ0 ]k(S n × S n ).
S
S
S
5
4-dimensional results
In this section we consider the exceptional case of a 4-dimensional p-stratifold
and give a similar classification result in that situation. The proof of the main
theorem can be found in §6.8.
S
For a 4-dimensional stratifold
, every link of the singularity Li is a 3dimensional manifold. According to the computation of Ω∗ by Thom [Th]
we immediately obtain from Theorem 1:
Algebraic & Geometric Topology, Volume 3 (2003)
1065
Resolutions of p-stratifolds with isolated singularities
Corollary 15 A four-dimensional p-stratifold with isolated singularities always admits a resolution.
If we further assume the links to be oriented we can use the following well-known
result, which can be proved easily.
Proposition 16 Every orientable 3-manifold is parallelizable, hence in particular spin.
The normal 1-type of a simply connected 4-dimensional spin-manifold is given
by BSpin. Since Ωspin
= 0 (see [Mi3, Lem. 9]) we obtain the following corollary
3
from Theorem 5.
Corollary 17 A four-dimensional p-stratifold with isolated singularities admits an optimal resolution if and only if all links of singularities are orientable.
We have to develop some notation in the topological category. Use BTOP to
denote the classifying space of topological vector bundles and let BTOPSpin be
the 2-connected cover over BTOP. Let M be a simply connected 4-manifold.
Using the Wu-Formula we can explain the Stiefel-Whitney-classes of M . We
call the topological manifold M spin if w2 (M ) vanishes. One can show that
the topological Gauss map of M lifts to BTOPSpin if and only if M is spin.
Note further that if such a lift exists, it is unique.
Using [Kr1, Thm. 2] and the h-cobordism-Theorem in dimension 4 [F, Thm.
10.3] we formulate:
Theorem 18 Let M1 and M2 be compact 4-dimensional topological spin
manifolds with e(M1 ) = e(M2 ) and let g : ∂M1 −→ ∂M2 be a homeomorphism compatible with the induced spin-structures on the boundaries. If
M1 ∪g M2 vanishes in ΩBTOPSpin
, then g can be extended to a homeomorphism
4
G : M1 ]k(S 2 × S 2 ) −→ M2 ]k(S 2 × S 2 ) for k ∈ {0, 1}.
We call two resolutions topologically equivalent if the diffeomorphism ϕc in the
definition of equivalent resolutions in §1 is replaced by a homeomorphism. Using
this notation, we obtain the following classification result in dimension four:
S
S
S
S
S
S
Theorem 19 Let ˆ −→
and ˆ0 −→
be two optimal resolutions of a
4-dimensional p-stratifold
with isolated singularities {xi }i∈I , such that each
link Li is connected. Assume that both ˆ and ˆ0 are spin and that for a
suitable representative U Σ of the neighbourhood germ, the following conditions
hold for all i ∈ I :
Algebraic & Geometric Topology, Volume 3 (2003)
S
1066
Anna Grinberg
• e(U Σ̂i ) = e(U Σ̂0i ),
• the spin-structures of U Σ̂i and U Σ̂0i coincide on the boundary,
• sign(U Σ̂i ∪∂ U Σ̂0i ) = 0.
S
S
Then ˆ]k(S 2 ×S 2 ) is topologically equivalent to ˆ0 ]k(S 2 ×S 2 ) for a k ∈ {0, 1}.
6
6.1
Outline of the proofs
Proof of Theorem 1
Although the proof can be found in [Kr3], it is useful to understand its nature
for the succeeding results.
One of the basic tools for constructing a resolving map is the following lemma,
which can be proved with the help of Morse theory, cf. [Kr3].
Lemma 20 Let W be a smooth compact manifold with boundary. Then there
is a codense compact subspace X of W and a continuous map f : ∂W −→ X
such that W is homeomorphic to ∂W × [0, 1] ∪f X , where on ∂W × [0, 1) the
homeomorphism can be chosen to be a diffeomorphism.
In other words, every smooth manifold with boundary arises from its collar by
attaching a codense set. The notation codense stands for the complement of a
dense subset. With this information we are ready to prove Theorem 1.
S
S
Proof Let p : ˆ −→
be a resolution. Set Wi := U Σ̂i . Since Wi is a
compact manifold with boundary Li , we obtain [Li ] = 0 in Ωn−1 .
Let on the other hand Wi be a compact manifold bounding Li and let f (N̊)
be the top stratum of
. Set ˆ := N ∪ (ti Wi ) and construct with the help
of the last lemma the following resolving map:
S
S
Sˆ ∼= N ∪i (∂Wi × [0, εi ] ∪f
i
Xi )
p
id
id
/
N ∪i (Li × [0, εi ] ∪ {xi }) ∼
=
6
Algebraic & Geometric Topology, Volume 3 (2003)
6
S.
6
1067
Resolutions of p-stratifolds with isolated singularities
6.2
Proof of Theorem 9
We consider a 2n-dimensional manifold M , which is (n − 1)-connected, and
want to determine its (n − 1) type. We begin with the classification up to
fiber homotopy equivalence of fibrations p : B −→ BO, with a CW-complex
B , fulfilling
(1) B is (n − 1)-connected and
(2) πi (F ) = 0 for i ≥ n, where F is the fiber.
Compare such a fibration with the (n − 1)-connected cover of BO:
BOhn − 1i
_
o
?
t
Fhn−1i
:
B
t
t
t
t
phn−1i
/
p
BO
Since all obstructions vanish, we obtain a lift p̄ : B −→ BOhn − 1i, which
without loss of generality may be assumed to be a fibration. From the long exact
homotopy sequence we see that the homotopy groups of the fiber vanish, except
in dimension (n − 1), where the group is π := coker(p∗ : πn (B) −→ πn (BO)).
Thus, p̄ : B −→ BOhn − 1i is a fibration with fiber K(π, n − 1). Such fibrations
are classified in [Ba, §5.2] as follows:
∼
=
[BOhn − 1i, K(π, n)]/[Aut(π)] −→ F(K(π, n − 1), BOhn − 1i)
[f ]
7−→
f ∗ (P (K(π, n))
Here F(K(π, n − 1), BOhn − 1i) denotes the set of all fibrations over BOhn − 1i
with fiber K(π, n − 1) up to fiber homotopy equivalence. Thus p̄ : B −→
BOhn − 1i is a pull back
B
/
P (K(π, n))
p̄
BOhn − 1i
/
θ
K(π, n)
with an appropriate map θ . The definition of π force the induced map θ∗ :
πn (BOhn − 1i) −→ πn (K(π, n)) = π to be surjective. Therefore we can assume
θ∗ to be the canonical projection to the factor group π . On the other hand,
each factor group of πn (BO) leads to a fibration with the claimed properties.
We summarize this discussion in
Lemma 21 Fibrations with properties 1. and 2. are given by pk : Bk −→ BO
up to fiber homotopy equivalence (k ∈ N, 0 ≤ k < |πn (BO)|).
Algebraic & Geometric Topology, Volume 3 (2003)
1068
Anna Grinberg
Consider now the fibration p : Bk −→ BO and ask for a lift:
w
Bk
;
w
ν:M
w
w
w
p
/
BO
With the help of obstruction theory we see that such a lift exists if and only if
im (ν∗ : πn (M ) −→ πn (BO)) ≤ hkxi,
where πn (BO) = hxi. Combining this discussion with Lemma 21, the statement
of Theorem 9 follows immediately.
6.3
Surgery in the middle dimension
First we give a brief introduction in surgery, for more details compare [W2].
Surgery is a tool to eliminate homotopy classes in the category of manifolds.
Let M be a compact m-dimensional manifold. One starts with an embedding
f : S r × D m−r ,→ M̊ and define T := D r+1 × D m−r ∪f (M × I), where f is
considered as a map to M × 1. The corners of the manifold T can always be
straighten, according to [CF]. This construction is called attaching an (r + 1)handle and T the trace of a surgery via f .
The boundary of T is M ∪ (∂M × I)∪ M 0 and we call M 0 the result of a surgery
of index r + 1 via f . It is not difficult to see that T can also be viewed as the
trace of a surgery on M 0 via the obvious embedding of D r+1 × S m−r−1 into
M 0 , compare [Mi2].
Since we are working in the category of manifolds with B -structures we have
to ask, whether the result of surgery via an embedding f is equipped with a
B -structure. For general results see [Kr1]. In our situation we are only looking
at fibrations pk : Bk −→ B defined in §3.
Lemma 22 Let M be a manifold of dimension m ∈ {2n, 2n + 1} with Bk structure and f : S n × D m−n ,→ M an embedding. Then ν̄ : M −→ B extends
to a normal Bk -structure on T , the trace of the surgery via f .
Proof The embedding f : S n ×D m−n ,→ M induces a normal Bk -structure on
S n × D m−n denoted by f ∗ ν̄ . There is a unique (up to homotopy) Bk -structure
on Dn+1 × D m−n and we have to show that its restriction to S n × D m−n is
Algebraic & Geometric Topology, Volume 3 (2003)
1069
Resolutions of p-stratifolds with isolated singularities
f ∗ ν̄ . Let Fk be the fiber of pk : Bk −→ BO. From the long exact homotopy
sequence we see that the different Bk structures on S n × D m−n are classified
by πn (Fk ). But the construction of Bk implies that πn (Fk ) = 0, thus the
restriction on S n × D m−n coincides with the given structure and we obtain a
Bk -structure on T .
Lemma 23 Let M be an (n − 1)-connected 2n-dimensional manifold. Let
Hn (M ) have a free basis {λ1 , . . . , λr , µ1 , . . . , µr } with
• λi · λj = 0 for all i, j and
• λi · µj = δij for all i, j .
If further each embedded sphere representing a generator λi has a trivial normal
bundle, then by a finite sequence of surgeries the homology group Hn (M ) can
be eliminated.
Remark The last two lemmas imply that the condition in Theorem 10 is
sufficient.
Proof According to a result of Haefliger ([Hae]), every element of Hn (M ) can
be represented by an embedding S n ,→ M . Let λ := λr and ϕ0 : S n ,→ M
an embedding representing λ. Since λ is assumed to have a trivial normal
bundle, the embedding can be extended to ϕ : S n × D n ,→ M . Set M0 :=
M − ϕ((S n × D n )◦ ) and let M 0 denote the result of surgery via ϕ, i.e. M 0 =
M0 ∪ϕ (D n+1 × S n−1 ). Combine now the long exact homology sequences of
pairs (M, M0 ) and (M 0 , M0 ) and obtain the following commutative diagram:
Z LLL
∂˜0
0
/
LL
L17
→λ
LL
LL
L
&
Hn (M0 )
/
i∗
Hn (M )
·λ
/
Z
∂˜
/
Hn−1 (M0 )
/
0
Hn (M 0 )
0
λ·
The excision, together with the surjectivity of Hn (M ) −→ Z, implies M0 is
(n − 1)-connected, further we see Hn (M0 ) = hλ1 , . . . , λr , β1 , . . . , βr−1 i.
Algebraic & Geometric Topology, Volume 3 (2003)
1070
Anna Grinberg
Thus M 0 is (n − 1)-connected as well and
Hn (M 0 ) = hλ̄1 , . . . , λ̄r−1 , µ̄1 , . . . , µ̄r−1 i,
where the generators are given by λ̄i = λi +λZ and µ̄i = µi +λZ. We can always
deform the embedding of the generator λ̄i to M0 , such that it represents the
class λi + λr ∈ Hn (M0 ). Thus we conclude λ̄i · λ̄j = 0 und λ̄i · µ̄j = δij .
Since the intersection product λi · λr vanishes we obtain ν(λi + λr ) = ν(λi ) +
ν(λr ) (cf. [W1]). Now we proceed with the manifold M 0 and inductively obtain
the desired statement.
6.4
Surgery on odd-dimensional manifolds
Lemma 24 Let T be a bordism in Ω2n (Bk ) between a manifold M of index k
and a homotopy sphere S . Then T is bordant in Ω2n (Bk ) rel. boundary to T 0 ,
such that its homology groups Hn (T 0 ) and Hn+1 (T 0 ) are free and Hi (T 0 ) = 0
for i 6∈ {0, n, n + 1, 2n + 1}.
Proof According to Theorem 6, we can assume that T is (n − 1)-connected,
further the Universal Coefficient Theorem implies that Hn+1 (T ) is free and
Tor(Hn (T )) ∼
= Tor(Hn (T, M )). We will show that the torsion of Hn (T, M ) can
be eliminated by a finite sequence of surgeries.
From the long exact homology sequence of the pair (T, M ) we see, that every
torsion element α0 ∈ Hn (T, M ) comes from an element α ∈ Hn (T ). After
possible correction of α by an element of Hn (M ) ∼
= πn (Bk ) we achieve ν(α) =
0.
Let ϕ : S n × D n+1 ,→ T̊ be an embedding representing α. As in the previous
proof, we set T0 = T −ϕ((S n ×D n+1 )◦ ) and T 0 = T0 ∪(D n+1 ×S n ). We combine
now the exact triple sequences for (T, T0 , M ) and (T 0 , T0 , M ) to obtain:
Algebraic & Geometric Topology, Volume 3 (2003)
1071
Resolutions of p-stratifolds with isolated singularities
0
Hn+1 (T0 , M )
Hn+1 (T 0 , M )
β·
Z PPPP
d0
Hn+1 (T, M )
α·
/
Z KK
d
/
PP
PP17→α0
PP
PP
PP
(
Hn (T0 , M )
KK
KK17→β 0
KK
KK
K
/
Hn (T, M )
/
0
j∗0
Hn (T 0 , M )
%
0
The element β ∈ Hn (T 0 ) is given by the embedding ψ : D n+1 × S n ,→ T 0 and
corresponds to the homotopy class [ψ|0×S n ].
We consider the two cases, where α is free or a torsion element modulo
i∗ (Hn (∂T )), separately.
Case 1 α is primitive (mod i∗ (Hn (∂T ))).
α·
In this case the Poincaré duality implies that the map Hn+1 (T, M ) −→ Z is
surjective, therefore
∼ Hn (T, M )/< α0 >.
Hn (T 0 , M ) =
Hence the torsion group of Hn (T 0 , M ) has been reduced.
Case 2 α is torsion (mod i∗ (Hn (∂T ))).
α·
The map Hn+1 (T, M ) −→ Z is trivial now. Denote with o(x) the order of a
torsion element x. From the sequence above we see that o(α0 )d(1) ⊂ im d, thus
there exists a b0 ∈ Z such that
o(α0 )d0 (1) = b0 d(1).
(∗)
If b0 = 0, then the element β 0 , corresponding to α0 , has infinite order, and the
torsion rang again decreases.
Algebraic & Geometric Topology, Volume 3 (2003)
1072
Anna Grinberg
If b0 6= 0, then (ker j∗0 ) ⊂ Tor(Hn (T0 , M )), thus j∗0 is injective on the free part
of Hn (T0 , M ), therefore the element d0 (1) has infinite order and o(β 0 ) | |b0 |. We
need a finer case differentiation.
Claim If n is even and α a torsion element of order a in Hn (T ), then β ∈
Hn (T 0 ) is an element of infinite order.
Consider the pair sequences (T, T0 ) and (T 0 , T0 ) and obtain the following diagram
0
Hn+1 (T0 )
Hn+1 (T 0 )
Z MMM
d0
0
/
Hn+1 (T0 )
/
Hn+1 (T )
α·
/
Z HH
d
/
MM
M17
α
M→
MM
MM
&
Hn (T0 )
HH
HH17→β
HH
HH
/
Hn (T )
/
0
Hn (T 0 )
#
0
As in the previous case, there exists a b ∈ Z such that
a · d0 (1) = b · d(1) in Hn (T0 )
or equivalently after the identification of the generators
(ϕ0 )∗ (a · ([S n ] ⊗ 1) − b · (1 ⊗ [S n ])) = 0.
By computing the self intersection number of a · ([S n ] ⊗ 1) − b · (1 ⊗ [S n ]) we
obtain 2ab([S n ] ∗ ⊗[S n ]∗ ) ∈ im ((ϕ0 )∗ : H 2n (W0 ) −→ H 2n (S n × S n )). Thus
2ab = 0, and since a 6= 0 it follows b = 0. We conclude again that β is of
infinite order, as described previously.
Claim If n is even and α has infinite order, then the element α0 will be
eliminated.
Algebraic & Geometric Topology, Volume 3 (2003)
Resolutions of p-stratifolds with isolated singularities
1073
In this situation we have α = α̃ + i∗ (γ), where α̃ is a torsion element and
γ ∈ Hn (M ). The normal bundle map is given by
ν : Hn (T ) −→ πn−1 (SO(n + 1)) = πn−1 (SO).
If πn−1 (SO) = 0 or Z the fact that the map ν : Hn (T ) −→ πn−1 (SO) is
a homomorphism implies, that α̃ already has a trivial normal bundle, this
leads us in the situation of the last claim. It remains to study the situation
πn−1 (SO) = Z/2 with ν(α̃) = 1. Observe that without loss of generality we
α·
can assume i∗ (γ) to be primitive. Thus the map Hn+1 (T ) −→ Z from the
sequence of the last claim is surjective and we obtain
Hn (T 0 ) ∼
= Hn (T )/ < α > ⇒ Hn (T 0 , M ) ∼
= Hn (T, M )/ < α0 > .
Claim If n is odd then the torsion group Tor(Hn (T, M )) can be reduced.
The group πn (SOn+1 ) acts on the trivializations via
(ω, ϕ) 7−→ ψ : S n × D n+1 ,→ T
(x, y)
7→ ϕ(x, ω(x)y)
Note, that the change of trivialization does not affect the B -structure on S n ×
Dn+1 since the induced B -structures ϕ∗ ν̄ and ψ ∗ ν̄ differ by an element from
πn (Fk ) = 0.
Let y0 be the basis point of S n , then we see
ψ0 (x, y0 ) = ϕ0 (x, ω(x)(y0 )) = ϕ0 (x, pw(x)) = ϕ0 (id × pω)∆(x),
where the map p : SO(n + 1) −→ S n is the canonical fiber bundle and ∆ :
S n −→ S n × S n the diagonal. Denote by il : S n −→ S n × S n the inclusion in
the l-th component and let ι be the generator of πn (S n ), set ιl := (il )∗ (ι). We
pass to homotopy and use the fact that the map π∗ δ : πn+1 (S n+1 ) −→ πn (S n )
from the long exact homotopy sequence for p is given by multiplication with 2
[Ste]. Thus we compute
(ψ0 )∗ (ι1 ) =
=
=
=
(ϕ0 )∗ (id × pω)∗ ∆∗ (ι)
(ϕ0 )∗ (id × pω)∗ (ι1 + ι2 )
(ϕ0 )∗ (ι1 + 2k · ι2 )
(ϕ0 )∗ (ι1 ) + 2k · (ϕ0 )∗ (ι2 )
Since M is (n − 1)-connected we obtain the corresponding statement in homology:
(ψ0 )∗ ([S n ] ⊗ 1) = (ϕ0 )∗ ([S n ] ⊗ 1) + 2k · (ϕ0 )∗ (1 ⊗ [S n ])
Algebraic & Geometric Topology, Volume 3 (2003)
1074
Anna Grinberg
The equality (∗) now becomes
b · (ψ0 )∗ (1 ⊗ [S n ]) = a · ((ψ0 )∗ ([S n ] ⊗ 1) − 2k(ψ0 )∗ (1 ⊗ [S n ]))
a · (ψ0 )∗ ([S n ] ⊗ 1) = (b + 2ka)(ψ0 )∗ (1 ⊗ [S n ])
Since the case b + 2ka = 0 has already been treated, we assume b + 2ka 6= 0.
By choosing k appropriately we can achieve
o(β 0 ) ≤ o(α0 ).
Let p be a prime number such that (α0 )p 6= 0 in Hn (T, M ; Fp ). From the
analogous sequences with Fp -coefficients we conclude
Hn (T 0 , M ; Fp ) ∼
= Hn (T, M ; Fp )/< α0 >
= Hn (T0 , M ; Fp )/im d0 ∼
and with universal coefficient theorem
|Tor(Hn (T 0 , M ))| < |Tor(Hn (T, M ))|.
Combining all cases together we see, that a torsion element of Hn (T, M ) can
either be eliminated by a finite sequence of surgeries or can be replaced by an
element of infinite order. Inductively we obtain the desired statement.
6.5
Proof of Theorem 10
Proof We only have to prove that every manifold normally Bk bordant to a
homology sphere is elementary.
Let T be a bordism between M and a homology sphere S . According to
Theorem 6, we can without loss of generality assume that T is (n−1)-connected
and that the homology groups Hn (T ) and Hn+1 (T ) are free. The long exact
sequence of the pair (T, M ) together with Poincaré duality leads to the following
commutative diagram:
Hn+1 (T )
/
Hn+1 (T, M )
P.−D.
H n (T, M )
δ
/
Hn (M )
P.−D.
j∗
/
H n (T )
i∗
/
Hn (T )
P.−D.
i∗
j∗
/
Hn (T, M )
/
0
P.−D.
/
H n (M )
d
/
H n+1 (T, M )
From this we see
dim Hn (M ) = 2(dim ker i∗ ).
Thus ker i∗ is a direct summand of Hn (M ). It is not hard to verify, that ker i∗
is isotropic. Let now λ : S n ,→ M be a representative of an element of ker i∗ ,
then the map can be extended to λ̄ : Dn −→ T and with the help of the
Whitney-trick [Hae] this map can be assumed to be an embedding. Thus the
restriction of the normal bundle of Dn to the boundary gives us a trivialization
of the normal bundle of λ.
Algebraic & Geometric Topology, Volume 3 (2003)
1075
Resolutions of p-stratifolds with isolated singularities
6.6
Proof of Theorem 11
Proof Using the statement of Theorem 10 we see, that the conditions of Theorem 6 are fulfilled for the manifolds U Σ̂i ]S and U Σ̂0i for a homotopy sphere
S . Thus, the diffeomorphism on the boundary can be extended to the the diffeomorphism U Σ̂i ]S]k(S n × S n ) −→ U Σ̂0i ]k(S n × S n ) for k ∈ {0, 1}. From
the definition of resolutions and from construction the diffeomorphism on the
boundaries we see that the obtained diffeomorphisms extend in the obvious way
to a diffeomorphism
]S]k(S n × S n ) −→ ]k(S n × S n ) having the desired
properties.
S
S
It remains to show, that the conditions of the theorem are true for every pair
of representatives of the neighbourhood germs [U Σ̂i ] and [U Σ̂0i ], once we have
checked them on a single representative pair. If ci : Li × [0, εi /2] −→ N and
di : Li × [0, ε0i /2] −→ N with εi < ε0i are two representatives of the germ of
collars around Li , then there exists a δi > 0 such that ci coincides with d0i on
Li × [0, δi ]. We choose a diffeomorphism ηi : [0, εi /2] −→ [0, ε0i /2] with ηi ≡ id
on [0, δi ]. The map induces an isomorphism
U Σic
−→ U Σid
f (ci (x, t)) 7−→ f (di (x, η(t)))
being the identity on a small neighbourhood of xi ∈ Σ. This gives us a diffeomorphism between U Σ̂ic and U Σ̂id making the following diagram commutative:

∂U Σ̂ic
/
U Σ̂ic
∼
=
/
U Σ̂id
_
o
?
∂U Σ̂id
∼
=
∼
=
∂U Σ̂0 ic

/
U Σ̂0 ic
∼
=
/
U Σ̂0 id
_
o
?
∂U Σ̂0 id
This completes the proof.
6.7
Algebraic invariants
ad (2) Let (H, Λ, sν∗ ) be elementary and L = hλ1 , . . . λk i a Lagrangian with
sν∗ |L ≡ 0. Thus sign(Λ) = 0 and 0 = sν∗ (λi ) = Λhκsν∗ , λi i for all 1 ≤ i ≤ k .
Since L is maximal it follows that κsν∗ ∈ L and therefore sν∗ (κsν∗ ) = 0.
On the other hand let sign(Λ) = 0 and sν∗ (κsν∗ ) = 0. Choose a basis
{λ1 , . . . , λk , µ1 , . . . , µk } of H such that Λ(λi , λj ) = 0 and Λ(λi , µj ) = δij .
There is nothing to show if κsν∗ = 0. Otherwise we can without loss of generality assume that sν∗ (λi ) = 0 for all i > 1, since sν∗ is a homomorphism.
Algebraic & Geometric Topology, Volume 3 (2003)
1076
Anna Grinberg
Recall the equality sν∗ (v) = P
Λ(κsν∗ , v) ∀v ∈ H . Since κsν∗ ∈ H there are
k
ai , bi ∈ Z such that κsν∗ =
i=1 (ai λi + bi µi ). Consider a sub-Lagrangian
0
0
L := hλ2 , . . . , λk i. If κsν∗ 6∈ L , build L := hλ2 , . . . , λk , κ̃sν∗ i, where κ̃sν∗ is
a primitive element of H with κsν∗ ∈ hκ̃sν∗ i. This is a Lagrangian, satisfying
sν∗ |L ≡ 0. In the case of κsν∗ ∈ L0 , the coefficients a1 , b1 , . . . , bk have to be
zero, thus L = hλ1 , . . . , λk i is a Lagrangian with the desired property.
ad (3) The conditions are obviously necessary. To see that they are also sufficient choose a symplectic basis {λ1 , . . . , λk , µ1 , . . . , µk } of H . Sort the generators in the following way
for i ≤ s,
for i > s,
sν∗ (λi ) = sν∗ (µi ) = 1
sν∗ (λi ) = 0
where s is an integer between 0 and k . The assumption
Φ(H) =
k
X
sν∗ (λi )sν∗ (µi ) = 0
i=1
implies that s ≡ 0(mod 2). Construct a new basis {λ01 , . . . , µ0k } for H by the
substitution
λ02i−1 = λ2i−1 + λ2i ,
λ02i = µ2i−1 − µ2i ,
0
µ2i−1 = µ2i
µ02i = λ2i
for 2i ≤ s, and
λ0i = λi
µ0i = µi
for i > s. This new basis is again symplectic and satisfies the condition
sν∗ (λ01 ) = · · · = sν∗ (λ0k ) = 0.
6.8
Proof of Theorem 19
Proof As in the proof of Theorem 11 we conclude that it is enough to show
that the diffeomorphism on the boundary ∂U Σ̂i −→ ∂U Σ̂0i can be extended to a
homeomorphism on U Σ̂i ]k(S 2 × S 2 ) −→ U Σ̂0i ]k(S 2 × S 2 ). Since the resolutions
are optimal, the manifolds U Σ̂i and U Σ̂0i are 1-connected, hence M := U Σ̂i ∪∂
U Σ̂0i is again 1-connected. In order to apply Theorem 18 we have to show that
the closed 4-dimensional spin manifold with vanishing signature is bordant
to a homotopy sphere. Then we apply the topological 4-dimensional Poincaré
conjecture proved by Freedman [F, Thm. 1.6] and obtain the desired statement.
Algebraic & Geometric Topology, Volume 3 (2003)
Resolutions of p-stratifolds with isolated singularities
1077
We want to use surgery to prove that M is bordant to a homotopy sphere.
Since M is spin and sign(M ) = 0, there is a basis {λ1 , . . . , λk , µ1 , . . . , µk } of
H2 (M ) satisfying
Λ(λi , λj ) = 0 Λ(µi , µj ) = 0 Λ(λi , µj ) = δij .
We can not use the Haefliger’s embedding theorem in dimension 4, but according to [F, Thm. 3.1, 1.1] every generator λi is represented by a topological
embedding S 2 × D 2 ,→ M . Knowing this we can proceed in exactly the same
way as in the proof of Lemma 23, working in the category TOP.
Lemma 24 is still valid in dimension 4 as well as the arguments in §6.6.
To complete the proof observe that according to arguments from §6.6 it is
enough to check the conditions for a single representative of the neighbourhood
germ.
References
[AK] S. Akbulut, H. King, Topology of Real Algebraic Sets. L’Enseignement
Mathématique, 29 (1983), 221-261.
[Ba]
H. H. Baues, Obstruction Theory, on Homotopy Classification of Maps. Springer
Lectures Notes 628, 1977.
[BR]
R. Benedetti and Jean-Jacques Risler, Real Algebraic and Semi-Algebraic Sets.
Hermann, Éditeurs des science et des arts, 1990.
[BT]
T. Bröcker and T. tom Dieck, Representations of Compact Lie Groups. SpringerVerlag, 1985.
[CF]
P. Conner and E. E. Floyd, Differentiable Periodic Maps. Springer-Verlag, 1972.
[F]
M. H. Freedman, The Topology of Four-Dimensional Manifolds. J. Differ. Geom.
17 (1982), 357-453.
[H]
M. W. Hirsch, Differential Topology. Springer-Verlag, 1976.
[Hae] A. Haefliger, Differentiable Imbeddings. Bull. Amer. Soc. 67 (1961), 109-112.
[Hi]
H. Hironaka, Resolutions of Singularities of an Algebraic Variety over a Field of
Characteristic Zero: I,II. Ann. of Math. 79 (1964), 109 - 326.
[Ke]
M. A. Kervaire, Some Non-Stable Homotopy Groups of Lie Groups. Illinois J.
Math. Vol 4 (1960), 161-169.
[Kr1] M. Kreck, Surgery and Duality. Annals of Mathematics Vol. 149 (1999), 704-754.
[Kr2] M. Kreck, Differential Algebraic Topology. Preprint.
[Kr3] M. Kreck, Stratifolds. Preprint.
Algebraic & Geometric Topology, Volume 3 (2003)
1078
Anna Grinberg
[Mi1] J. Milnor, A Procedure for Killing the Homotopy Groups of Differentiable Manifolds. Symposia in Pure Math. A.M.S., Vol. III (1961), 39-55.
[Mi2] J. Milnor, Morse Theory. Annals of Math. Studies 51, Princeton University
Press, 1965.
[Mi3] J. Milnor, Remarks Concerning Spin Manifolds. Differ. and Combinat. Topology,
Sympos. Marston Morse, Princeton, 1965, 55-62.
[Mi4] J. Milnor, Singular Points of Complex Hypersurfaces. Princeton University
Press, 1968.
[St]
R. E. Stong, Notes on Cobordism Theory. Mathematical Notes, Princeton University Press, 1968.
[Ste]
N.Steenrod, The Topology of Fibre Bundles. Princeton University Press, 1951.
[Th]
R. Thom, Quelques Properiétés Globales des Varietétés Differentiables. Comm.
Mathe. Helv. Vol 28 (1954), 17-86.
[W1] C. T. C. Wall, Classification of (n − 1)-Connected 2n-Manifolds. Annals of
Mathematics Vol. 75 (1962), 163-189.
[W2] C. T. C. Wall, Surgery on Compact Manifolds. London Math. Soc. Monographs
1, Academic Press, New York, 1970.
Department of Mathematics, UC San Diego
9500 Gilman Drive, La Jolla, CA, 92093-0112
Email: agrinber@math.ucsd.edu
Received: 17 March 2003
Revised: 15 September 2003
Algebraic & Geometric Topology, Volume 3 (2003)